3rd international castle meeting on coding theory and ... · edgar mart´ınez-moro universidad de...

Joaquim Borges and Mercè Villanueva (eds.)

3rd International Castle Meeting on Coding Theory and Applications

Universitat Autònoma de BarcelonaServei de Publicacions

Bellaterra, 2011

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3rd International Castle Meeting on Coding Theory and Applications : [September 11-15, 2011, Cardona, Spain, organized by the Combinatorics, Coding and Security Group, Universitat Autònoma de Barcelona] / Joaquim Borges and Mercè Villanueva (eds.)— Bellaterra (Barcelona) : Universitat Autònoma de Barcelona. Servei de Publicacions, 2011. — (Congressos de la Universitat Autònoma de Barcelona; 5)

ISBN 978-84-490-2688-1

I. Borges, J. (Joaquim); Villanueva, M. (Mercè)III. Col·lecció1. Criptografia – Informàtica – Congressos2. Codificació, Teoria de la – Congressos519.71

Edició:Universitat Autònoma de BarcelonaServei de PublicacionsEdifici A. 08193 Bellaterra (Cerdanyola del Vallès). SpainTel. 93 581 10 22Fax 93 581 32 [email protected]/publicacions

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ISBN 978-84-490-2688-1Dipòsit legal: B.29.077-2011

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3rd International Castle Meeting on Coding Theory and ApplicationsJ. Borges and M. Villanueva (eds.)

5

3ICMCTASeptember 11-15, 2011, Cardona, Spain

organized by the Combinatorics, Coding and Security GroupUniversitat Autonoma de Barcelona

Steering Committee

Angela I. Barbero University of Valladolid, SpainØyvind Ytrehus University of Bergen, Norway

Organizing Committee

Joaquim Borges, UAB Co-chair of the Scientific CommitteeCristina Fernandez, UAB Co-chair of the Local CommitteeJaume Pujol, UAB Co-chair of the Local CommitteeJosep Rifa, UAB Chair of the Steering CommitteeMerce Villanueva, UAB Co-chair of the Scientific Committee

Local Committee

Muhammad Bilal UAB, SpainBernat Gaston UAB, SpainJaume Pernas UAB, SpainMarta Pujol UAB, SpainLorena Ronquillo UAB, Spain

Sponsoring Institutions

Catalan Research Agency AGAURIEEE Information Theory SocietySARE-Consolider Ingenio 2010 ProjectSpanish Ministry of Science and InnovationUniversitat Autonoma de Barcelona

6Organization

Scientific Committee

Peter Beelen Technical University of Denmark, DenmarkJuergen Bierbrauer Michigan Technological University, USAMaria Bras-Amoros Universitat Rovira i Virgili, SpainClaude Carlet University of Paris VIII, FranceGerard Cohen Telecom ParisTech, FranceItalo Dejter University of Puerto Rico, USAAlexandros Dimakis University of Southern California, USASteven Dougherty University of Scranton, USAIwan Duursma University of Illinois, USACristina Fragouli Ecole polytechnique Federale de Lausanne, SwitzerlandHeeralal Janwa University of Puerto Rico, USASan Ling Nanyang Technological University, SingaporeEdgar Martınez-Moro Universidad de Valladolid, SpainPatric Ostergard Aalto University, FinlandRuud Pellikaan Eindhoven University of Technology, NetherlandsKevin T. Phelps Auburn University, USAAngel del Rıo Universidad de Murcia, SpainEirik Rosnes University of Bergen, NorwayPatrick Sole Telecom ParisTech, FranceEmina Soljanin Bell Labs New Jersey, USAFaina I. Solov’eva Sobolev Institute of Mathematics, RussiaJos Weber Delft University of Technology, NetherlandsVictor A. Zinoviev Institute for Problems of Information Transmission, Russia

External Reviewers

Ezio Biglieri Antoine LobsteinStanislav Bulygin Carmen MartınezPierre-Louis Cayrel Carlos MunueraLara Dolecek Irene Marquez CorbellaMark Flanagan Frederique OggierRyan Gabrys Carla RafolsIiro Honkala Bernardo G. RodriguesCarmelo Interlando Leo StormeDenis S. Krotov Horacio Tapia Recillas


7

Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Invited speakers

Managing Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Robert Calderbank

Error-correcting Codes in the Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Tuvi Etzion

On Old and New Results in Algebraic Coding Theory over Ring Alphabets . . . . . . . . 19Marcus Greferath

Codes from Incidence Matrices of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Jennifer D. Key

Communications

A Generalization of Lee Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Carlos Araujo, Italo Dejter, and Peter Horak

Sphere Coverings and Identifying Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31David Auger, Gerard Cohen, and Sihem Mesnager

Induced Perfect Colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Sergey V. Avgustinovich and Ivan Yu. Mogilnykh

A Proof of the MDS Conjecture over Prime Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Simeon Ball

Information Sets in Abelian Codes: Defining Sets and Groebner Basis . . . . . . . . . . . . . 45Jose Joaquın Bernal and Juan Jacobo Simon

One Class of Generalized Quasi-cyclic (L,G) Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 51Sergey Bezzateev and Natalia Shekhunova

Binary Self-dual Codes from 3-class Association Schemes . . . . . . . . . . . . . . . . . . . . . . 57Muhammad Bilal, Joaquim Borges, Steven T. Dougherty, and CristinaFernandez-Cordoba

On Properties of Propelinear and Transitive Binary Codes . . . . . . . . . . . . . . . . . . . . . . 63Joaquim Borges, Josep Rifa, and Faina I. Solov’eva

Two New Families of Binary Completely Regular Codes and Distance Regular Graphs 69Joaquim Borges, Josep Rifa, and Victor A. Zinoviev

8Table of contents

On the Geil-Matsumoto Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Maria Bras-Amoros and Albert Vico-Oton

Identifying Codes over L-graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Cristobal Camarero, Carmen Martınez, and Ramon Beivide

A Construction of MDS Array Codes Based on Companion Matrices . . . . . . . . . . . . . 85Sara D. Cardell, Joan-Josep Climent, and Veronica Requena

Maximum Distance Separable 2D Convolutional Codes of Rate 1/n . . . . . . . . . . . . . 91Joan-Josep Climent, Diego Napp, Carmen Perea, and Raquel Pinto

Pure Asymmetric Quantum MDS Codes from CSS Construction . . . . . . . . . . . . . . . . . 97Martianus Frederic Ezerman, Somphong Jitman, and San Ling

Classification of Linear Codes with Prescribed Minimum Distance and New UpperBounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Thomas Feulner

Sub-optimum Soft-input Decoding of OMEC Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Jurgen Freudenberger and Farhad Ghaboussi

Evaluation Codes Defined by Finite Families of Plane Valuations at Infinity . . . . . . . . 115Carlos Galindo and Francisco Monserrat

Group Codes which are not Abelian Group Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Cristina Garcıa Pillado, Santos Gonzalez, Victor Markov, Consuelo Martınez,and Alexandr Nechaev

Quasi-cyclic Minimum Bandwidth Regenerating Codes . . . . . . . . . . . . . . . . . . . . . . . . 127Bernat Gaston, Jaume Pujol, and Merce Villanueva

Coding Strategies for Reliable Storage in Multi-level Flash Memories . . . . . . . . . . . . . 133Kathryn Haymaker and Christine A. Kelley

A Note on Linear Programming Based Communication Receivers . . . . . . . . . . . . . . . . 141Shunsuke Horii, Tota Suko, Toshiyasu Matsushima, and Shigeichi Hirasawa

On MDS Convolutional Codes of Rate 1/n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Jose Ignacio Iglesias Curto, Jose Marıa Munoz Porras, Francisco Javier PlazaMartın, and Gloria Serrano Sotelo

Bounds on Collaborative Decoding of Interleaved Hermitian Codes with a DivisionAlgorithm and Virtual Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Sabine Kampf

Improving Security of Niederreiter Type GPT Cryptosystem Based on ReducibleRank Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Eraj Khan, Ernst M. Gabidulin, Bahram Honary, and Hassan Ahmed


9

On the Automorphism Groups of the Additive 1-perfect Binary Codes . . . . . . . . . . . . 169Denis S. Krotov

Enumeration of Spectrum Shaped Binary Run-length Constrained Sequences . . . . . . . 175Oleg F. Kurmaev

A Note on (xvt, xvt−1)-minihypers in PG(t, q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Ivan Landjev and Peter Vandendriessche

A Matrix Approach to Group Convolutional Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Sergio R. Lopez-Permouth and Steve Szabo

Classification of the Extremal Self-dual Codes with 2-transitive Automorphism Groups 193Anton Malevich and Wolfgang Willems

Evaluation of Public-key Cryptosystems Based on Algebraic Geometry Codes . . . . . 199Irene Marquez-Corbella, Edgar Martınez-Moro, and Ruud Pellikaan

A Class of Repeated-root Abelian Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205Edgar Martınez-Moro, Hakan Ozadam, Ferruh Ozbudak, and Steve Szabo

On the Permutation Automorphism Group of Quaternary Linear Hadamard Codes . . . 211Jaume Pernas, Jaume Pujol, and Merce Villanueva

Minimal Realizations of a 2-D Convolutional Code of Rate 1/n by SeparableRoesser Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Telma Pinho, Raquel Pinto, and Paula Rocha

On the Subfield Subcodes of Hermitian Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Fernando L. Pinero and Heeralal Janwa

On the Multidimensional Permanent and q-ary Designs . . . . . . . . . . . . . . . . . . . . . . . . 231Vladimir N. Potapov

Geometrically Uniform Quasi-perfect Codes Derived from Graphs over Integer Rings 237Catia Regina de Oliveira Quilles Queiroz and Reginaldo Palazzo Junior

GPT Cryptosystem for a Random Network Coding Channel . . . . . . . . . . . . . . . . . . . . . 243Haitham Rashwan, Nina I. Pilipchuk, Ernst M. Gabidulin, and Bahram Honary

On the Capacity of a Discretized Gaussian Shift Channel . . . . . . . . . . . . . . . . . . . . . . . 251Eirik Rosnes, Angela I. Barbero, Guang Yang, and Øyvind Ytrehus

Codes Associated with Circulant Graphs and Permutation Decoding . . . . . . . . . . . . . . 257Padmapani Seneviratne

Fast Skew-feedback Shift-register Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265Vladimir Sidorenko and Martin Bossert

A Generalisation of the Gilbert-Varshamov Bound and its Asymptotic Evaluation . . . 271Ludo Tolhuizen

10Table of contents

New Binary Codes From Extended Goppa Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277Martin Tomlinson, Mubarak Jibril, Cen Tjhai, Markus Grassl, and MohammedZaki

Discrete Logarithm Like Problems and Linear Recurring Sequences . . . . . . . . . . . . . . 283Hugo Villafane, Santos Gonzalez, Llorenc Huguet, and Consuelo Martınez

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289


11

Preface

The III International Castle Meeting on Coding Theory and Applications has been held atCardona Castle in Catalonia (Spain) on September 11-15, 2011. It was organized by theresearch group CCSG (Combinatorics, Coding and Security Group) from the UniversitatAutonoma de Barcelona. The main objectives of the conference were the communicationof scientific and technological results, the cooperation among research groups at an interna-tional level, and the promotion of young pre-doc and post-doc researchers on the topics ofthe conference.

The present Proceedings contain the extended abstracts of 4 invited talks and 43 com-munications. The previous review process assures the high quality of these works. It is re-markable the international character of the conference, since there were participants, invitedspeakers, steering committee, organizing committee, scientific committee and local commit-tee members from 22 different countries.

The organizing committee thanks to all for their contribution, specifically, to the steeringand scientific committees, as well as to the people out of these committees helping in thereviewing process, to the 4 invited speakers and to all the participants. Also, the conferencehas been possible thanks to the financial support of the following institutions: IEEE Informa-tion Theory Society, Spanish Ministry of Science and Innovation, Catalan Research AgencyAGAUR, and Universitat Autonoma de Barcelona.

We are also grateful with the journal Designs, Codes and Cryptography for accepting topublish a full version of the more excellent papers presented at the conference.

September 2011 Joaquim BorgesMerce Villanueva

Co-chairs of the Scientific Committee


13

Managing Interference

Robert Calderbank

[email protected]

Duke University, USA

1 Extended abstract

We consider a framework for full-duplex communication in ad-hoc wireless networks re-cently proposed by Dongning Guo. An individual node in the wireless network either trans-mits or it listens to transmissions from other nodes but it cannot to both at the same time.There might be as many nodes as there are 48 bit MAC addresses but we assume that onlya small subset of nodes contribute to the superposition received at any given node in thenetwork.

We use ideas from compressed sensing to show that simultaneous communication is pos-sible across the entire network. Our approach is to manage interference through configurationrather than to eliminate or align it through extensive exchange of fine-grained Channel StateInformation. Our approach to configuration makes use of old fashioned coding theory.


15

Error-correcting Codes in the Projective Space?

Tuvi Etzion

[email protected]

Technion — Israel Institute of Technology, Israel

1 Extended abstract

Let Fq be the finite field of order q, and letW be an arbitrary (fixed) vector space of dimensionn over Fq . SinceW is isomorphic to Fnq , we can assume thatW is in fact Fnq . The projectivespace of order n over Fq , denoted herein by Pq(n), is the set of all the subspaces of Fnq ,including 0 and Fnq itself. Given a nonnegative integer k ≤ n, the set of all subspaces ofFnq that have dimension k is known as a Grassmannian, and usually denoted by Gq(n, k).Thus Pq(n) = ∪0≤k≤nGq(n, k). It is well known that

|Gq(n, k)| =[nk

]q

def=

(qn − 1)(qn−1 − 1) · · · (qn−k+1 − 1)

(qk − 1)(qk−1 − 1) · · · (q − 1)

where[nk

]q

is the q-ary Gaussian coefficient. It turns out that the natural measure of distance,

the subspace distance in Pq(n), is given by

dS(X,Y )def= dim(X) + dim(Y )− 2 dim(X ∩ Y )

for all X,Y ∈ Pq(n). It is well known (cf. [1,2]) that the function above is a metric; thusboth Pq(n) and Gq(n, k) can be regarded as metric spaces. Given a metric space, one candefine codes. We say that C ⊆ Pq(n) is an (n,M, d) code in projective space if |C| = Mand dS(X,Y ) ≥ d for all X,Y ∈ C. If an (n,M, d) code C is contained in Gq(n, k) forsome k, we say that C is an (n,M, d, k) code. An (n,M, d, k) code is also called a constantdimension code.

The (n,M, d), respectively (n,M, d, k), codes in projective space are akin to the familiarcodes in the Hamming space, respectively (constant weight) codes in the Johnson space,where the Hamming distance serves as the metric. There are, however, important differences.For all q, n and k, the metric space Gq(n, k) corresponds to a distance-regular graph, similar? This research was supported in part by the United States — Israel Binational Science Foundation

(BSF), Jerusalem, Israel, under Grant 2006097.


17

Finally, some more results, new related references, and a list of open problems for furtherresearch are given.

References

1. R. Ahlswede, H. K. Aydinian, and L. H. Khachatrian, On perfect codes and related concepts, De-signs, Codes, and Cryptography, 22 (2001), 221–237.

2. R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEETrans. Inform. Theory, IT-54 (2008), 3579–3591.

3. L. Chihara, On the zeros of the Askey-Wilson polynomials, with applications to coding theory,SIAM J. Math. Anal., 18 (1987), 191–207.

4. W. J. Martin and X. J. Zhu, Anticodes for the Grassman and bilinear forms graphs, Designs, Codes,and Cryptography, 6 (1995), 73–79.

5. M. Schwartz and T. Etzion, Codes and anticodes in the Grassman graph, Journal of CombinatorialTheory, Series A, 97 (2002), 27–42.

6. T. Etzion and A. Vardy, Error-correcting codes in projective space, in Proc. ISIT 2008, Toronto(2008), 871–875.

7. T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory,IT-57 (2011), 1165–1173.

8. A. Kohnert and S. Kurz, Construction of large constant dimensiona codes with a prescribed mini-mum distance, Lecture Notes in Computer Science, 5393 (2008), 31–42.

9. D. Silva, F. R. Kschischang, and R. Koetter A Rank-metric approach to error control in randomnetwork coding, IEEE Trans. Inform. Theory, Series A, IT-54 (2008), 3951–3967.

10. S. T. Xia and F. W. Fu, Johnson type bounds on constant dimension codes, Designs, Codes, Designs,Codes, and Cryptography, 50 (2009), 163–172. IEEE Trans. Inform. Theory, IT-56 (2010), 3207–3216.

11. M. Gadouleau and Z. Yan, Packing and covering properties of subspace codes for error control inrandom linear network coding, IEEE Trans. Inform. Theory, IT-56 (2010), 2097–2108.

12. M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes,IEEE Trans. Inform. Theory, IT-56 (2010), 3207–3216.

13. P. Frankl and R. M. Wilson, The Erdos-Ko-Rado theorem for vector spaces, Journal of Combina-torial Theory, Series A, 43 (1986), 228–236.

14. S. Thomas, Designs over finite fields, Geom. Dedicata, 21 (1987), 237–242.15. S. Thomas, Designs and partial geometries over finite fields, Geom. Dedicata, 63 (1996), 247–253.16. H. Suzuki, 2-designs over GF(2m), Graphs Combin., 6 (1990), 293–296.17. H. Suzuki, 2-designs over GF(q), Graphs Combin, 8 (1992), 381–389.18. M. Miyakawa, A. Munemasa and S. Yoshiara, On a class of small 2-designs over GF(q), Journal

Combinatorial Designs, 3 (1995), 61–77.19. T. Itoh, A new family of 2-designs over GF(q) admitting SLm(q`), Geom. Dedicata, 69 (1998),

261–286.20. H. Wang, C. Xing, and R. M. Safavi-Naini, Linear authentication codes: bounds and constructions,

IEEE Trans. Inform. Theory, IT-49 (2003), 866–872.21. M. Braun, A. Kerber and R. Laue, Systematic construction of q-analogs of t-(v, k, λ)-designs,

Designs, Codes, and Cryptography, 34 (2005), 55–70.22. C. Bachoc, Semidefinite programming, harmonic analysis, and coding theory, in Lecture Notes

CIMPA Summer School on Semidefinite Programming in Algebraic Combinatorics, (2009), 1–47.23. T. Etzion and N. Silberstein, Error-correcting codes in projective space via rank-metric codes and

Ferrers diagrams, IEEE Trans. Inform. Theory, IT-55 (2009), 2909–2919.24. P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, Journal of Com-

binatorial Theory, Series A, 25 (1978), 226–241.

18 Error-correcting Codes in the Projective SpaceTuvi Etzion

25. E. M. Gabidulin, Theory of codes with maximum rank distance, Problems of Information Trans-mission, 21 (1985), 1–12.

26. R. M. Roth, Maximum-rank array codes and their application to crisscross error correction, IEEETrans. Inform. Theory, IT-37 (1991), 328–336.

27. A.-L. Trautmann and J. Rosenthal, New improvements on the echelon-Ferrers construction, inProc. Int. Symp. on Math. Theory of Networks and Systems, (2010), 405–408.

28. N. Silberstein and T. Etzion, Large constant dimension codes and lexicodes, Advances in Mathe-matics of Communications, 5 (2011), 177-189.

29. N. Silberstein and T. Etzion, Codes and designs related to lifted MRD codes, to be presented inProc. ISIT 2011, Saint Petersburg (2011).

30. T. Etzion and A. Vardy, On q-Analogs for Steiner Systems and Covering Designs, Advances inMathematics of Communications, 5 (2011), 161-176.


19

On Old and New Results in Algebraic CodingTheory over Ring Alphabets

Marcus Greferath

[email protected]

University College Dublin, Ireland

1 Extended abstract

Ring-linear algebraic coding theory gained importance during the last decade of the previouscentury, when it was discovered that certain non-linear binary codes of high quality can bebetter understood as linear codes over the ring of integers modulo 4.

Since then, a number of workgroups worldwide have been doing research in this newdiscipline of Applicable Algebra. Their results suggest that most of the foundational questionsof algebraic coding over rings have been settled by now, whereas strong examples of record-breaking codes are still in demand.

This talk gives some insight into this amazingly beautiful chapter of Discrete Mathema-tics. We will report on a collection of results from the literature and from our own previousand current research. The talk will finish with open problems and projects for future research.


21

Codes from Incidence Matrices of Graphs

Jennifer D. Key

[email protected]

University of the Western Cape, South Africa

1 Introduction

An incidence matrix for an undirected graph Γ = (V,E) is a |V | × |E| matrix G = [gX,e]with rows labelled by the vertices X ∈ V and columns by the edges e ∈ E, where gX,e = 1if X ∈ e, gX,e = 0 if X 6∈ e.

For any prime p let Cp(G) denote the row span ofG over Fp from a graph Γ . In a numberof recent papers, for example [5,4,7,12,13], the codes Cp(G) for some classes of regularconnected graphs were studied. It was found that for these classes the codes have parameters

[|E|, |V | − εp, δ(Γ )]p

where ε2 = 1, εp = 0, 1 for p odd, δ(Γ ) is the minimum degree of Γ , and the words ofminimum weight are precisely the non-zero scalar multiples of the rows ofG of weight δ(Γ ).In particular when Γ is k-regular and so δ(Γ ) = k, this implies that in these cases the graphcan be retrieved from the code. Furthermore, in some of the classes, these properties led tosimilar facts for the binary codes of the adjacency matrices of the associated line graphs,these being subcodes of the binary codes from the incidence matrices of the original graphs.Indeed, it was a study of the codes from the adjacency matrices of triangular graphs in [11]that pointed to this focus on the incidence matrices.

In addition, it was noticed that the weight enumerator of the code of the incidence matrixhad, in all cases studied, a gap between the weight k for the valency, and 2k − 2 for thedifference of two rows, i.e. the valency of the line graph. This then immediately shows that,in these cases, the binary code of an adjacency matrix of the line graph of a graph Γ has theproperty that the minimum weight is either the valency of Γ or the valency of the line graph;in the latter case, that the words of that weight are the rows of the adjacency matrix might notnecessarily follow, but does in fact seem to be true in most of the classes studied.

The question was thus asked whether these properties are in fact general for graphs sat-isfying certain conditions. We make a start at answering this question here by using the con-cept of edge-connectivity to show that this is indeed the case for many classes of graphs. Weoutline this method and some of the results obtained in Section 3, but first we mention, inSection 2 below, the classes studied that led to this observation.

22 Codes from Incidence Matrices of GraphsJennifer D. Key

2 Classes of graphs studied previously

Infinite classes of graphs studied and found, by combinatorial and coding theoretic methods,along with induction, to have the properties described for Cp(G), G an incidence matrix,include:

– Hamming graphs Hk(n,m) (see [5,4])The Hamming graph Hk(n,m), for n, k,m integers, 1 ≤ k < n, is the graph withvertices themn n-tuples ofRn, whereR is a set of sizem, and adjacency defined by twon-tuples being adjacent if they differ in k coordinate positions. They are the graphs fromthe Hamming association scheme. In particular, the n-cube: Qn = H(n, 2) = H1(n, 2)(R = F2).

– Uniform subset graphs Γ (n, k,m)

A uniform subset graph Γ (n, k,m) has vertex set Ωk, where |Ω| = n, and adjacencydefined by a ∼ b if |a ∩ b| = m. The symmetric group Sn always acts on these graphs.All classes studied satisfy the properties described, and include:

• the odd graphs Γ (2k + 1, k, 0) (see [2])• triangular graphs Γ (n, 2, 1) (strongly regular) and Γ (n, 2, 0) (see [6])• Γ (n, 3,m) for m = 0, 1, 2. (see [3])

– Complete multipartite graphs Kn1,n2,...,nk

• Kn the complete graph (see [12])• Kn,n the complete bipartite graph (see [13,14])• Kn,m for n 6= m

• Kn1,n2,...,nk where ni = n for i = 1, . . . , k

– Strongly regular graphs (n, k, λ, µ)

A k-regular graph Γ = (V,E) with |V | = n is strongly regular with parameters (n, k, λ,µ) if

• for any P,Q ∈ V such that P ∼ Q, |R ∈ V | R ∼ P &R ∼ Q| = λ, and• for any P,Q ∈ V such that P 6∼ Q, |R ∈ V | R ∼ P &R ∼ Q| = µ.

The classes found to have the described property include:

• Triangular graphs T (n) = L(Kn), (line graph of the complete graph, also a uniformsubset graph), n ≥ 4, (

(n2

), 2(n− 2), n− 2, 4) (see [12])

• Paley graphs P (q), vertex set Fq where q ≡ 1 (mod 4) and x ∼ y if x − y is anon-zero square, (q, q−12 , q−54 , q−14 ) (see [7])

• Lattice graphs L2(n) = L(Kn,n), the line graph of the complete bipartite graph,(n2, 2(n− 1), n− 2, 2) (see [14])

• Symplectic graphs (see [10]):Γ2m(q) with parameters ( q

2m−1q−1 , q

2m−1−1q−1 − 1, q

2m−2−1q−1 − 2, q

2m−2−1q−1 )

and complementΓ c2m(q) with parameters ( q

2m−1q−1 , q2m−1, q2m−2(q − 1), q2m−2(q − 1))

where m ≥ 2, q a prime power.

3rd international castle meeting on coding theory and ... · edgar mart´ınez-moro universidad de...

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